arXiv subjects
Publications and source records attributed to Alejandro Santacruz Hidalgo.
Given a von Neumann algebra with a semifinite normal faithful trace, the notion of a core decreasing operator is introduced. This construction depends on a family of commuting projections called an ordered core. With these tools in place, a version of the level function of a measurable operator is introduced. The down space of a strongly symmetric space of $τ$-measurable operators is defined.
Boundedness of an abstract formulation of Hardy operators between Lebesgue spaces over general measure spaces is studied and, when the domain is L^1, shown to be equivalent to the existence of a Hardy inequality on the half line with general Borel measures. This is done by extending the greatest decreasing minorant construction to general measure spaces depending on a totally ordered collection of measurable sets, called an ordered core. A functional description of the greatest decreasing minorant is given, and for a large class of ordered cores, a pointwise description is provided. As an application, characterizations of Hardy inequalities for metric measure spaces are given, we note that the metric measure space is not required to admit a polar decomposition.
Given a measure space and a totally ordered ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the Köthe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves. The down spaces of $L^1$ and $L^\infty$ are shown to form an exact Calderón couple with divisibility constant $1$; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderón couple with divisibility constant $1$; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.